Summary

Important Concepts

A vector is a quantity described by both a magnitude and a direction.

The figure illustrates the characteristics of a displacement vector labeled "A".

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Unit Vectors

Unit vectors have magnitude 1 and no units. Unit vectors $î$ and $ĵ$ define the directions of the x- and y-axes.

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Using Vectors

Components

The component vectors are parallel to the x- and y-axes:

\vec{A} = \vec{A}_{x} + \vec{A}_{y} = A_{x} î + A_{y} ĵ

In the figure at the right, for example:

\begin{array}{l} A_{x} = A cos θ A = \sqrt{A_{x}^{2} + A_{y}^{2}} \\ A_{y} = A sin θ θ = t a n^{- 1} (A_{y} / A_{x}) \end{array}

▶ Minus signs need to be included if the vector points down or left.

The figure illustrates the components of a displacement vector labeled vector A using the x y coordinate plane and its value in the different quadrants.

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The components $A_{x}$ and $A_{y}$ are the magnitudes of the component vectors $\vec{A}_{x}$ and $\vec{A}_{y}$ and a plus or minus sign to show whether the component vector points toward the positive end or the negative end of the axis.

Working Graphically

The figure consists of five parts that illustrate the addition of vectors using the tip to tail rule and the parallelogram rule method, negative of a vector, subtraction of vectors, and multiplication of vectors.

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Working Algebraically

Vector calculations are done component by component: $\vec{C} = 2 \vec{A}_{y} + \vec{B} means {\begin{matrix} C_{x} = 2 A_{x} + B_{x} \\ C_{y} = 2 A_{y} + B_{y} \end{matrix}$

The magnitude of $\vec{C}$ is then $C = \sqrt{C_{x}^{2} + C_{y}^{2}}$ and its direction is found using ${tan}^{- 1}$ .